Implicit Plots

Functions implicitly defined by an equation may be plotted using a relatively efficient algorithm of linear interpolation upon triangles. As described in the 2D-plot window Insert > Plot commandB2C2._, the form of an implicitly defined equation is u = v, where u and v are expressions of at most two variables. Examples of such equations are

2 2
x + y = 4

3 2
x = y - 5

x = 3

and

2
r + cos(θ) = 0

To produce an implicit plot in rectangular coordinates, the variables in the equation are assigned to the horizontal and vertical axes. Usually, the primary variable is assigned to the horizontal axis, the secondary variable to the vertical axis (see Ordering Variables in ExpressionsOrdering_Variables_in_Expressions). If the equation contains only one variable, it will normally be assigned to the vertical axis. However, if the variable is x, as in the third equation above, x will be assigned to the horizontal axis. In general, if an equation contains the variable x, the equation will always be plotted with x assigned to the horizontal axis. Thus, for implicit plotting in rectangular coordinates we recommend using x for the horizontal and y for the vertical coordinate.

To produce an implicit plot in polar coordinates, the angle and the distance from the origin are assigned to variables in the equation. Usually, the primary variable is assigned to the angle, the secondary variable to the distance from the origin. However, if one of the variables in the equation is a Greek letter, it is assigned to the angle. For implicit plotting in polar mode we recommend using r for the distance from the origin and θ for the angle.

The implicit plotting algorithm starts in the top left corner of the 2D-plot window, plotting points in a top to bottom, left to right fashion. Unlike most explicit plots, implicit plots may not be automatically scaled; nor may an implicit plot line be traced.

Explicit plots are faster and more accurate than implicit plots. Before plotting, when possible, Derive automatically converts a function implicitly defined by an equation into a function explicitly defined by an equation of the form y = u where y is a variable and u is a univariate expression in another variable. For example, the linear equation

2x + 3y = 6

is transformed into the equation

y = —2/3x + 2

and then plotted using the more expeditious explicit plotting algorithm.

A function implicitly defined by an equation may be solved for one of its variables in terms of the other by using the algebra window’s Solve > Expression command1PET_XA to generate explicit solutions. If explicit solutions of the equation are achievable, explicit plots of these solutions may be superimposed in a 2D-plot window.

An alternate method for generating explicit solutions for implicitly defined equations is to substitute polar coordinates for the variables and solve for the distance from the origin in terms of the angle. For example,

5 4 5 4
x + x = y + y

cannot be solved exactly for either x or y. However, substituting r cos(θ) for x and r sin(θ) for y, then solving for r gives the two explicit solutions

r = 0

and

2
1 - 2 COS(θ)
r = ———————————————————
5 5
COS(θ) - SIN(θ)

These explicit solutions may then be plotted in polar coordinates in the 2D-plot window.

Implicit 2D plots of a family of plot lines can be used to make contour plots of functions of two variables. In a 2D-plot window after issuing the Options > Simplify Before Plotting command49ZWMQ, plotting an expression of the form

VECTOR (z = u, z, m, n, s)

produces a contour plot of the function z = u where u is a function of x and y as z varies from m to n in steps of size s. For example, the function

z = 10/(2^2+(x-2)^2+y^2) + 10/(2^2+(x+2)^2+y^2)

plots as "two mountains" in a 3D-plot window; whereas the vector

VECTOR(z = 10/(2^2+(x-2)^2+y^2) + 10/(2^2+(x+2)^2+y^2), z, 1, 3, 0.1)

plots as a contour plot of the function in a 2D-plot window. Note that the Calculus > Vector command.WK1F5 is an easy way to enter such expressions.